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Theorem 0pledm 19567
Description: Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
0pledm.1  |-  ( ph  ->  A  C_  CC )
0pledm.2  |-  ( ph  ->  F  Fn  A )
Assertion
Ref Expression
0pledm  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  ( A  X.  { 0 } )  o R  <_  F
) )

Proof of Theorem 0pledm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0pledm.1 . . . 4  |-  ( ph  ->  A  C_  CC )
2 sseqin2 3562 . . . 4  |-  ( A 
C_  CC  <->  ( CC  i^i  A )  =  A )
31, 2sylib 190 . . 3  |-  ( ph  ->  ( CC  i^i  A
)  =  A )
43raleqdv 2912 . 2  |-  ( ph  ->  ( A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
5 0cn 9086 . . . . . 6  |-  0  e.  CC
6 fnconstg 5633 . . . . . 6  |-  ( 0  e.  CC  ->  ( CC  X.  { 0 } )  Fn  CC )
75, 6ax-mp 8 . . . . 5  |-  ( CC 
X.  { 0 } )  Fn  CC
8 df-0p 19564 . . . . . 6  |-  0 p  =  ( CC  X.  { 0 } )
98fneq1i 5541 . . . . 5  |-  ( 0 p  Fn  CC  <->  ( CC  X.  { 0 } )  Fn  CC )
107, 9mpbir 202 . . . 4  |-  0 p  Fn  CC
1110a1i 11 . . 3  |-  ( ph  ->  0 p  Fn  CC )
12 0pledm.2 . . 3  |-  ( ph  ->  F  Fn  A )
13 cnex 9073 . . . 4  |-  CC  e.  _V
1413a1i 11 . . 3  |-  ( ph  ->  CC  e.  _V )
15 ssexg 4351 . . . 4  |-  ( ( A  C_  CC  /\  CC  e.  _V )  ->  A  e.  _V )
161, 13, 15sylancl 645 . . 3  |-  ( ph  ->  A  e.  _V )
17 eqid 2438 . . 3  |-  ( CC 
i^i  A )  =  ( CC  i^i  A
)
18 0pval 19565 . . . 4  |-  ( x  e.  CC  ->  (
0 p `  x
)  =  0 )
1918adantl 454 . . 3  |-  ( (
ph  /\  x  e.  CC )  ->  ( 0 p `  x )  =  0 )
20 eqidd 2439 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
2111, 12, 14, 16, 17, 19, 20ofrfval 6315 . 2  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  A. x  e.  ( CC  i^i  A
) 0  <_  ( F `  x )
) )
22 fnconstg 5633 . . . . 5  |-  ( 0  e.  CC  ->  ( A  X.  { 0 } )  Fn  A )
235, 22ax-mp 8 . . . 4  |-  ( A  X.  { 0 } )  Fn  A
2423a1i 11 . . 3  |-  ( ph  ->  ( A  X.  {
0 } )  Fn  A )
25 inidm 3552 . . 3  |-  ( A  i^i  A )  =  A
26 c0ex 9087 . . . . 5  |-  0  e.  _V
2726fvconst2 5949 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2827adantl 454 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2924, 12, 16, 16, 25, 28, 20ofrfval 6315 . 2  |-  ( ph  ->  ( ( A  X.  { 0 } )  o R  <_  F  <->  A. x  e.  A  0  <_  ( F `  x ) ) )
304, 21, 293bitr4d 278 1  |-  ( ph  ->  ( 0 p  o R  <_  F  <->  ( A  X.  { 0 } )  o R  <_  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   {csn 3816   class class class wbr 4214    X. cxp 4878    Fn wfn 5451   ` cfv 5456    o Rcofr 6306   CCcc 8990   0cc0 8992    <_ cle 9123   0 pc0p 19563
This theorem is referenced by:  xrge0f  19625  itg20  19631  itg2const  19634  i1fibl  19701  itgitg1  19702  ftc1anclem5  26286  ftc1anclem7  26288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-cnex 9048  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-mulcl 9054  ax-i2m1 9060
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ofr 6308  df-0p 19564
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